A100878 -id:A100878 - cp's OEIS frontend

A061336 Smallest number of triangular numbers which sum to n.

0, 1, 2, 1, 2, 3, 1, 2, 3, 2, 1, 2, 2, 2, 3, 1, 2, 3, 2, 3, 2, 1, 2, 3, 2, 2, 3, 2, 1, 2, 2, 2, 3, 3, 2, 3, 1, 2, 2, 2, 3, 3, 2, 2, 3, 1, 2, 3, 2, 2, 3, 2, 3, 3, 3, 1, 2, 2, 2, 3, 2, 2, 3, 3, 2, 2, 1, 2, 3, 2, 2, 3, 2, 2, 3, 3, 2, 3, 1, 2, 3, 2, 3, 2, 2, 3, 3, 2, 2, 3, 2, 1, 2, 2, 2, 3, 3, 2, 3, 2, 2, 2, 2, 3, 3

Offset: 0

Henry Bottomley, Apr 25 2001

nonn

a(n)=3 if n=5 or 8 mod 9, since triangular numbers are {0,1,3,6} mod 9.
From Bernard Schott, Jul 16 2022: (Start)
In September 1636, Fermat, in a letter to Mersenne, made the statement that every number is a sum of at most three triangular numbers. This was proved by Gauss, who noted this event in his diary on July 10 1796 with the notation:
EYPHKA! num = DELTA + DELTA + DELTA (where Y is in fact the Greek letter Upsilon and DELTA is the Greek letter of that name).
This proof was published in his book Disquisitiones Arithmeticae, Leipzig, 1801. (End)

			a(3)=1 since 3=3, a(4)=2 since 4=1+3, a(5)=3 since 5=1+1+3, with 1 and 3 being triangular.

Elena Deza and Michel Marie Deza, Fermat's polygonal number theorem, Figurate numbers, World Scientific Publishing (2012), Chapter 5, pp. 313-377.
C. F. Gauss, Disquisitiones Arithmeticae, Yale University Press, 1966, New Haven and London, p. 342, art. 293.

Giovanni Resta, Table of n, a(n) for n = 0..10000
George E. Andrews, EYPHKA! num = Delta + Delta + Delta, J. Number Theory 23 (1986), 285-293.
Eric Weisstein's World of Mathematics, Fermat's Polygonal Number Theorem.

Cf. A000217, A007294, A057945, A061337, A051533, A020757.

Cf. A100878 (analog for A000326), A104246 (analog for A000292), A283365 (analog for A000332), A283370 (analog for A000389).

t[n_]:=n*(n+1)/2; a[0]=0; a[n_]:=Block[ {k=1, tt= t/@ Range[Sqrt[2*n]]}, Off[IntegerPartitions::take]; While[{} == IntegerPartitions[n, {k}, tt, 1], k++]; k]; a/@ Range[0, 104] (* Giovanni Resta, Jun 09 2015 *)

\\ see A283370 for generic code, working but not optimized for this case of triangular numbers. - M. F. Hasler, Mar 06 2017

a(n)=my(m=n%9,f); if(m==5 || m==8, return(3)); f=factor(4*n+1); for(i=1,#f~, if(f[i,2]%2 && f[i,1]%4==3, return(3))); if(ispolygonal(n,3), n>0, 2) \\ Charles R Greathouse IV, Mar 17 2022

a(n) = 0 if n=0, otherwise 1 if n is in A000217, otherwise 2 if n is in A051533, otherwise 3 in which case n is in A020757.

a(n) <= 3 (proposed by Fermat and proved by Gauss). - Bernard Schott, Jul 16 2022

A338494 Least number of pentagonal pyramidal numbers needed to represent n.

Original entry on oeis.org

1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 7, 3, 4, 5, 6, 7, 8, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 7, 3, 4, 3, 4, 5, 6, 2, 3, 4, 5, 6, 7, 3, 4, 5, 6, 7, 8, 4, 5, 4, 5, 6, 1, 2, 3, 4, 5, 2, 2, 3, 4, 5, 6, 3, 3, 4, 5, 5, 6, 4, 2, 3, 4, 5, 6, 3, 3, 4, 5

Offset: 1

Ilya Gutkovskiy, Oct 30 2020

nonn

Eric Weisstein's World of Mathematics, Pentagonal Pyramidal Number
Index to sequences related to pyramidal numbers

Cf. A002411, A100878, A104246, A338491, A338493, A338495.

A338480 Least number of heptagonal numbers needed to represent n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 2, 3, 4, 5, 1, 2, 3, 3, 4, 5, 6, 2, 3, 4, 4, 5, 6, 7, 3, 4, 1, 2, 2, 3, 4, 4, 5, 2, 3, 3, 4, 5, 5, 6, 3, 4, 4, 5, 2, 3, 3, 1, 2, 3, 4, 3, 4, 4, 2, 3, 4, 5, 4, 5, 2, 3, 3, 4, 4, 2, 3, 3, 4, 4, 5, 5, 3, 1, 2, 3, 4, 5, 3, 4, 2, 2, 3, 3, 4, 4, 5, 3, 3, 4, 4, 2, 3, 4

Offset: 1

Ilya Gutkovskiy, Oct 29 2020

nonn

Eric Weisstein's World of Mathematics, Heptagonal Number
Index to sequences related to polygonal numbers

Cf. A000566, A002828, A061336, A100878, A338479, A338481.

A338479 Least number of hexagonal numbers needed to represent n.

Original entry on oeis.org

1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 2, 3, 4, 1, 2, 3, 3, 4, 5, 2, 3, 4, 4, 5, 6, 3, 1, 2, 2, 3, 4, 4, 2, 3, 3, 4, 5, 5, 3, 4, 4, 2, 3, 1, 2, 3, 4, 3, 4, 2, 3, 4, 5, 4, 2, 3, 3, 4, 2, 3, 3, 4, 4, 5, 1, 2, 3, 4, 5, 3, 2, 2, 3, 3, 4, 4, 3, 3, 4, 2, 3, 4, 3, 4, 4, 3, 3, 4, 2, 1, 2, 3, 2, 3, 3, 2, 3, 4, 3, 3

Offset: 1

Ilya Gutkovskiy, Oct 29 2020

nonn

Eric Weisstein's World of Mathematics, Hexagonal Number
Index to sequences related to polygonal numbers

Cf. A000384, A002828, A061336, A100878, A234533, A338480, A338481.

A338481 Least number of octagonal numbers needed to represent n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 2, 3, 4, 5, 6, 1, 2, 3, 3, 4, 5, 6, 7, 2, 3, 4, 4, 5, 6, 7, 8, 3, 4, 5, 1, 2, 2, 3, 4, 4, 5, 6, 2, 3, 3, 4, 5, 5, 6, 7, 3, 4, 4, 5, 6, 2, 3, 3, 4, 1, 2, 3, 4, 3, 4, 4, 5, 2, 3, 4, 5, 4, 5, 5, 2, 3, 3, 4, 4, 5, 2, 3, 3, 4, 4, 5, 5, 6, 3, 4, 1, 2, 3, 4, 5, 3

Offset: 1

Ilya Gutkovskiy, Oct 29 2020

nonn

Eric Weisstein's World of Mathematics, Octagonal Number
Index to sequences related to polygonal numbers

Cf. A000567, A002828, A061336, A100878, A338479, A338480.

A133929 Positive integers that cannot be expressed using four pentagonal numbers.

Original entry on oeis.org

9, 21, 31, 43, 55, 89

Offset: 1

Eric W. Weisstein, Sep 29 2007

Equivalently, integers m such that the smallest number of pentagonal numbers (A000326) which sum to m is exactly five, that is, A100878(a(n)) = 5. Richard Blecksmith & John Selfridge found these six integers among the first million, they believe that they have found them all (Richard K. Guy reference). - Bernard Schott, Jul 22 2022

			   9 =  5 +  1 + 1 + 1 + 1.
  21 =  5 +  5 + 5 + 5 + 1.
  31 = 12 + 12 + 5 + 1 + 1.
  43 = 35 +  5 + 1 + 1 + 1.
  55 = 51 +  1 + 1 + 1 + 1.
  89 = 70 + 12 + 5 + 1 + 1.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section D3, Figurate numbers, pp. 222-228.

Eric Weisstein's World of Mathematics, Pentagonal Number

Cf. A000326, A007527, A100878.

Equals A003679 \ A355660.

A338491 Least number of centered pentagonal numbers needed to represent n.

Original entry on oeis.org

1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 7, 3, 4, 5, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 7, 3, 4, 5, 6, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 7, 3, 4, 5, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 7, 3, 4, 5, 6, 2, 3, 4, 5, 6, 7, 3, 4, 5, 6

Offset: 1

Ilya Gutkovskiy, Oct 30 2020

nonn

Eric Weisstein's World of Mathematics, Centered Pentagonal Number
Index entries for sequences related to centered polygonal numbers

Cf. A005891, A100878, A101412, A234533, A338482, A338484, A338492, A338494.

A355660 Numbers m such that the smallest number of pentagonal numbers (A000326) which sum to m is exactly 4.

Original entry on oeis.org

4, 8, 16, 19, 20, 26, 30, 33, 38, 42, 50, 54, 60, 65, 67, 77, 81, 84, 88, 90, 96, 99, 100, 101, 111, 112, 113, 120, 125, 131, 135, 138, 142, 154, 159, 160, 166, 170, 171, 183, 195, 204, 205, 207, 217, 224, 225, 226, 229, 230, 236, 240, 241, 243, 255, 265, 275, 277, 286, 306, 308, 345

Offset: 1

Bernard Schott, Jul 12 2022

nonn

Richard Blecksmith & John Selfridge found 204 such integers among the first million, the largest of which is 33066. They believe that they have found them all (Richard K. Guy reference).

a(205) > 10^11, if it exists, from Giovanni Resta in A003679.

			4 = 1 + 1 + 1 + 1.
8 = 5 + 1 + 1 + 1.
16 = 5 + 5 + 5 + 1.
Also, it is not possible to get these terms when summing three or fewer pentagonal numbers.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section D3, Figurate numbers, pp. 222-228.

Richard K. Guy, Every number is expressible as the sum of how many polygonal numbers?, Amer. Math. Monthly 101 (1994), 169-172.

Cf. A000326, A100878.

Equals A003679 \ A133929.

nn = 100;
pen = Table[n (3n - 1)/2, {n, 0, nn - 1}];
lst = Range[pen[[-1]]];
Do[n = pen[[i]]+pen[[j]]+pen[[k]]; If[n <= pen[[-1]], lst = DeleteCases[lst, n]], {i, 1, nn}, {j, i, nn}, {k, j, nn}];
A003679 = lst;
Complement[A003679, {9, 21, 31, 43, 55, 89}] (* Jean-François Alcover, Jul 13 2022, after T. D. Noe in A003679 *)

A100878(a(n)) = 4.

A355717 Smallest number of generalized pentagonal numbers which sum to n.

Original entry on oeis.org

0, 1, 1, 2, 2, 1, 2, 1, 2, 2, 2, 3, 1, 2, 2, 1, 2, 2, 3, 2, 2, 3, 1, 2, 2, 3, 1, 2, 2, 2, 2, 2, 3, 2, 2, 1, 2, 2, 2, 3, 1, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 1, 2, 2, 3, 2, 2, 1, 2, 2, 3, 2, 2, 2, 2, 3, 2, 3, 3, 2, 1, 2, 2, 2, 3, 2, 3, 1, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 3, 2, 3, 2, 1, 2, 2, 3, 2, 2, 3, 2, 1

Offset: 0

Bernard Schott, Jul 15 2022

nonn

From Euler's Pentagonal Number Theorem, every number is expressible as the sum of at most three generalized pentagonal numbers (A001318) (see Richard K. Guy reference).
Corresponding sums of only pentagonal numbers of positive rank are A100878(n). Those numbers are a subset of the generalized pentagonals so that a(n) <= A100878(n).
More specifically, by the definition given in the name, we understand the following: Given n >= 0 we seek a multiset S such that (1) S is a multiset of GPN = {0, 1, 2, 5, ...} = A001318; (2) Sum(S) = n; (3) if T is a multiset of GPN and Sum(T) = n then card(T) >= card(S). Additionally one might require that the set is not empty. If a multiset satisfies these three conditions, then a(n) = card(S). Note that no actual summation has to be performed to decide the value of a(n); only membership in GPN needs to be tested, as shown in the Maple and Python program. - Peter Luschny, Jul 18 2022

			Let GPN = {0, 1, 2, 5, ...} be the generalized pentagonal numbers.
a(0) = 0 since {} is a multiset of GPN, Sum {} = 0, and card({}) = 0.
a(1) = 1 since {1} is a multiset of GPN, Sum {1} = 1, and card({1}) = 1.
a(3) = 2 since {1, 2} is a multiset of GPN, Sum {1, 2} = 3, and card({1, 2}) = 2.
a(11) = 3 since {2, 2, 7} is a multiset of GPN, Sum {2, 2, 7} = 11, card({2, 2, 7}) = 3, and no other multiset S of GPN with Sum(S) = 11 has less members.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section D3, Figurate numbers, pp. 222-228.

Eric Weisstein's World of Mathematics, Pentagonal Number Theorem.

Cf. A001318, A093519 (indices of 3's).

Cf. A100878.

A355717_list := proc(upto) local P, Q, k, q, isgpn; P := []; Q := [0];
isgpn := k -> ormap(n -> 0 = 8*k - (n + irem(n,2)) * (3*n + 2 - irem(n,2)), [$0..k]);
for k from 1 to upto do
    q := 3;
    if isgpn(k) then
        P := [op(P), k]; q := 1;
        elif ormap(p -> member(k - p, P), P) then q := 2 fi:
        Q := [op(Q), q];
od: Q end:
print(A355717_list(100));  # Peter Luschny, Jul 18 2022

def A355717_list(ln: int) -> list[int]:
    P: list[int] = []
    Q: list[int] = [0]
    def is_gpn(k: int) -> bool:
        return any(8 * k == ((n + n % 2) * (3 * n + 2 - n % 2)) for n in range(k + 1))
    for k in range(1, ln):
        q = 3
        if is_gpn(k):
            P.append(k)
            q = 1
        elif any([(k - p) in P for p in P]):
            q = 2
        Q.append(q)
    return Q
print(A355717_list(100))  # Peter Luschny, Jul 18 2022

a(n) <= 3.

a(A001318(n)) = 1.

A355774 An extension of the generalized pentagonal numbers such that every positive integer can be represented as the sum of at most two terms of the sequence.

Original entry on oeis.org

0, 1, 2, 5, 7, 11, 12, 15, 21, 22, 25, 26, 35, 39, 40, 49, 51, 57, 67, 70, 77, 87, 92, 100, 117, 120, 123, 126, 145, 153, 155, 173, 176, 182, 186, 187, 205, 210, 214, 222, 228, 241, 247, 251, 260, 283, 287, 301, 319, 330, 345, 376, 382, 392, 425, 435, 442, 448

Offset: 0

Peter Luschny, Jul 17 2022

nonn

The sequence is defined inductively. Starting from the empty sequence, the terms are added one after the other. A term is added if it is a generalized pentagonal number or if it cannot be represented as the sum of two preceding terms. Note that these exceptions form a proper subsequence of A093519.

Thus any positive number can be expressed as the sum of at most two positive terms by Euler's Pentagonal Number Theorem. Every pentagonal number and every generalized pentagonal number is in this sequence.

			32 = 7 + 25; 195 = 22 + 173.

Andreas Enge, William Hart and Fredrik Johansson, Short addition sequences for theta functions, arXiv:1608.06810 [math.NT], 2016.
Burkard Polster (Mathloger), The hardest 'What comes next?' (Euler's pentagonal formula), YouTube video, 2020.

Cf. A000326, A001318, A093519, A100878, A355717, A176747 (same construction with triangular numbers).

A355774_list := proc(upto) local P, k, issum, isgpn; P := [];
isgpn := k -> ormap(n -> 0 = 8*k-(n+irem(n,2))*(3*n+2-irem(n,2)), [$0..k]);
issum := k -> ormap(p -> member(k - p, P), P);
for k from 0 to upto do
    if isgpn(k) or not issum(k) then P := [op(P), k] fi od;
P end: print(A355774_list(448));

isgpn[k_] := AnyTrue[Range[0, k], 0 == 8*k-(#+Mod[#,2])*(3*#+2-Mod[#,2])&];
issum[k_] := AnyTrue[P, MemberQ[P, k-#]&];
P = {};
For[k = 0, k <= 448, k++, If[isgpn[k] || !issum[k], AppendTo[P, k]]];
P (* Jean-François Alcover, Mar 07 2024, after Peter Luschny *)

def A355774_list(upto: int) -> list[int]:
    P: list[int] = []
    for k in range(upto + 1):
        if any(
            k == ((n + n % 2) * (3 * n + 2 - n % 2)) >> 3
            for n in range(k + 1)
        ) or not any([(k - p) in P for p in P]):
            P.append(k)
    return P
print(A355774_list(448))

cp's OEIS Frontend

A061336 Smallest number of triangular numbers which sum to n.

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A338494 Least number of pentagonal pyramidal numbers needed to represent n.

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A338480 Least number of heptagonal numbers needed to represent n.

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A338479 Least number of hexagonal numbers needed to represent n.

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A338481 Least number of octagonal numbers needed to represent n.

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A133929 Positive integers that cannot be expressed using four pentagonal numbers.

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A338491 Least number of centered pentagonal numbers needed to represent n.

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A355660 Numbers m such that the smallest number of pentagonal numbers (A000326) which sum to m is exactly 4.

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A355717 Smallest number of generalized pentagonal numbers which sum to n.

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A355774 An extension of the generalized pentagonal numbers such that every positive integer can be represented as the sum of at most two terms of the sequence.